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Theorem tfrcldm 6354
Description: Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.)
Hypotheses
Ref Expression
tfrcl.f  |-  F  = recs ( G )
tfrcl.g  |-  ( ph  ->  Fun  G )
tfrcl.x  |-  ( ph  ->  Ord  X )
tfrcl.ex  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
tfrcl.u  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
tfrcl.yx  |-  ( ph  ->  Y  e.  U. X
)
Assertion
Ref Expression
tfrcldm  |-  ( ph  ->  Y  e.  dom  F
)
Distinct variable groups:    f, G, x    S, f, x    f, X, x    f, Y, x    ph, f, x
Allowed substitution hints:    F( x, f)

Proof of Theorem tfrcldm
Dummy variables  z  a  b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrcl.yx . . 3  |-  ( ph  ->  Y  e.  U. X
)
2 eluni 3808 . . 3  |-  ( Y  e.  U. X  <->  E. z
( Y  e.  z  /\  z  e.  X
) )
31, 2sylib 122 . 2  |-  ( ph  ->  E. z ( Y  e.  z  /\  z  e.  X ) )
4 tfrcl.f . . . 4  |-  F  = recs ( G )
5 tfrcl.g . . . . 5  |-  ( ph  ->  Fun  G )
65adantr 276 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Fun  G )
7 tfrcl.x . . . . 5  |-  ( ph  ->  Ord  X )
87adantr 276 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Ord  X )
9 tfrcl.ex . . . . 5  |-  ( (
ph  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
1093adant1r 1231 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  X  /\  f : x --> S )  ->  ( G `  f )  e.  S )
11 feq2 5341 . . . . . . . 8  |-  ( a  =  x  ->  (
f : a --> S  <-> 
f : x --> S ) )
12 raleq 2670 . . . . . . . 8  |-  ( a  =  x  ->  ( A. b  e.  a 
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
1311, 12anbi12d 473 . . . . . . 7  |-  ( a  =  x  ->  (
( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <-> 
( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) ) )
1413cbvrexv 2702 . . . . . 6  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b ) ) ) )
15 fveq2 5507 . . . . . . . . . 10  |-  ( b  =  y  ->  (
f `  b )  =  ( f `  y ) )
16 reseq2 4895 . . . . . . . . . . 11  |-  ( b  =  y  ->  (
f  |`  b )  =  ( f  |`  y
) )
1716fveq2d 5511 . . . . . . . . . 10  |-  ( b  =  y  ->  ( G `  ( f  |`  b ) )  =  ( G `  (
f  |`  y ) ) )
1815, 17eqeq12d 2190 . . . . . . . . 9  |-  ( b  =  y  ->  (
( f `  b
)  =  ( G `
 ( f  |`  b ) )  <->  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
1918cbvralv 2701 . . . . . . . 8  |-  ( A. b  e.  x  (
f `  b )  =  ( G `  ( f  |`  b
) )  <->  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )
2019anbi2i 457 . . . . . . 7  |-  ( ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2120rexbii 2482 . . . . . 6  |-  ( E. x  e.  X  ( f : x --> S  /\  A. b  e.  x  ( f `  b )  =  ( G `  ( f  |`  b
) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) )
2214, 21bitri 184 . . . . 5  |-  ( E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  ( f `  b )  =  ( G `  ( f  |`  b ) ) )  <->  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) )
2322abbii 2291 . . . 4  |-  { f  |  E. a  e.  X  ( f : a --> S  /\  A. b  e.  a  (
f `  b )  =  ( G `  ( f  |`  b
) ) ) }  =  { f  |  E. x  e.  X  ( f : x --> S  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) ) }
24 tfrcl.u . . . . 5  |-  ( (
ph  /\  x  e.  U. X )  ->  suc  x  e.  X )
2524adantlr 477 . . . 4  |-  ( ( ( ph  /\  ( Y  e.  z  /\  z  e.  X )
)  /\  x  e.  U. X )  ->  suc  x  e.  X )
26 simprr 531 . . . 4  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  e.  X )
274, 6, 8, 10, 23, 25, 26tfrcllemres 6353 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  -> 
z  C_  dom  F )
28 simprl 529 . . 3  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  z )
2927, 28sseldd 3154 . 2  |-  ( (
ph  /\  ( Y  e.  z  /\  z  e.  X ) )  ->  Y  e.  dom  F )
303, 29exlimddv 1896 1  |-  ( ph  ->  Y  e.  dom  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353   E.wex 1490    e. wcel 2146   {cab 2161   A.wral 2453   E.wrex 2454   U.cuni 3805   Ord word 4356   suc csuc 4359   dom cdm 4620    |` cres 4622   Fun wfun 5202   -->wf 5204   ` cfv 5208  recscrecs 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-recs 6296
This theorem is referenced by:  tfrcl  6355  frecfcllem  6395  frecsuclem  6397
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